Final answer:
To find the equation of the line tangent to the circle at a point where the x value is 3, find the derivative of the circle equation, substitute the x coordinate into the derivative equation, solve for y', and substitute the values into the equation y - y₁ = m(x - x₁).
Step-by-step explanation:
To find the equation of the line tangent to the circle at a point where the x value is 3, we need to find the derivative of the equation of the circle. The derivative represents the slope of the tangent line.
The equation of the circle is x² + y² + 10x - 8y - 59 = 0. We can rewrite this as (x + 5)² + (y - 4)² = 90, which is in the standard form of a circle equation: (x - h)² + (y - k)² = r².
The derivative of the equation is 2(x + 5) + 2(y - 4)y' = 0. Now we can substitute the x coordinate, which is 3, into the derivative equation to find the slope at that point.
2(3 + 5) + 2(y - 4)y' = 0
16 + 2(y - 4)y' = 0
From here, we can solve for y' and substitute the values back into the equation y - y₁ = m(x - x₁) to find the equation of the tangent line.